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MasTec was created out of a merger of two southeast utility firms, Church & Tower and Burnup & Sims. Mas Canosa purchased Church and Tower in 1971, having arriving almost penniless from Cuba in 1960.

He found work as a milkman, stevedore and shoe salesman while raising money for anti-Castro groups. Eventually he was named branch manager for the infrastructure company that became Church and Tower. He later bought the firm with the help of a $50,000 loan.

A contract to lay cable for the Southern Bell telephone company set the company on its current trajectory. As computer networking became commonplace, the company experienced rapid growth. It went public in 1994 as MasTec.

"Thinking about the struggle so many Cuban Americans went through, so many people in this community who left their homeland in search of a better life for their families, who made a lot sacrifices, we wake up every day striving to make that legacy proud," said Mas, Jorge's youngest son. "It’s humbling to think that we're now one of the 500 largest companies in the U.S., it’s pretty awesome."

Last year, MasTec reported $6 billion in revenues. It currently has a market cap of nearly $4 billion based on its share price of about $52. MasTec shares have climbed 17 percent over the past 12 months. The company just won a $500 million contract to help rebuild Puerto Rico's hurricane-damaged electrical grid. Jose and his brother Jorge are part of the group bidding alongside David Beckham to bring an MLS team to South Florida.

Mas said that thanks to ongoing strong demand in the energy and telecom sectors, he is confident that MasTec will be a Fortune 500 mainstay moving forward. Those divisions of MasTec drew more than a half billion dollars in revenue each in the first three months of the year, according to the company's most recent quarterly filing. The company now employs more than 22,000 workers.

"We’re blessed in that our end markets are in a really good spot," he said. "There's always a lot to do in the background to make this technology possible, to make it effective and efficient."

Mas believes Miami's status as an immigrant hub has made it an entrepreneurial hotspot. While it took MasTec a few decades to make the Fortune 500, Mas doubts his company will be the last Miami-based company to make the list.

"When somebody comes to this country in search of a better life, they’re hungrier, and that hunger makes the way they view business challenges differently," he said. "It's a special attribute that's especially present in Miami, and I think you have a greater chance of success because of it. You have no choice but to succeed."

Mastec was one of five Miami-Dade and Broward companies to make this year's 500. The others were Miami-Dade-based World Fuel Services, which markets and finances commercial fuel; homebuilder Lennar and logistics provider Ryder; and Broward-based auto-dealer group Autonation.